Solve the heat equation $$\frac{\partial u}{\partial{t}}=\nu\frac{\partial^2{u}}{\partial{x}^2},t>0,x>0$$ where $u(x,0)=f(t),u(x,0)=u_0$ using Laplace transforms.
My trouble is dealing with the general initial condition to find constants. Applying the Laplace transform I got $$U(s)=A(s)e^{\sqrt{\frac{s}{\nu}}x}+B(s)e^{-\sqrt{\frac{s}{\nu}}x}.$$ The issue is finding out $A$ and $B$. Using $u(0,s)=F(s)$, I get $$A(s)+B(s)+\frac{u_0}{s}=F(s).$$ The only way I can think of determining them is by forcing $U$ is bounded. Then $A=0$ and I can write $B(s)=F(s)-\frac{u_0}{s}$ and the solution is $$u(t,x)=L^{-1}(F(s)e^{\sqrt{\frac{s}{\nu}}x})-u_0\text{erfc}(\frac{x}{2\sqrt{\nu t}})+u_0.$$ Is this the way to proceed?